Rasiak struck with a firm header from Inigo Idiakez's corner in the 16th minute after the Rams had battered the relegation-haunted Cheshire side early on.The Pole's header flew into the top corner, but Smith should have extended their lead on several occasions before the break.
The Crewe defence stood off him time and time again as he came close with shots from the edge of the box.
Marcus Tudgay too should have done better after the craftsmanship of Idiakez had prised another opening, but he shot narrowly wide on the turn.
Idiakez curled a free- kick just over Ben Williams' bar before Crewe finally offered a threat to the rampant Rams when Luke Varney flicked Kenny Lunt's cross straight at Lee Camp.
The Derby keeper had more to do after the interval and had to get behind Varney's rising drive as Crewe forced their way back into contention.
But Williams still needed to make a smart parry to keep out Idiakez's dangerous drive from finding the bottom corner.
A dreadful backpass by Chris McCready gifted Smith an opening, but the striker took the wrong option and Williams was able to close him off by the by-line and make the block.
Crewe appealed in vain for a penalty in the 59th minute but it seemed Michael Johnson's last-ditch tackle on Steve Jones had been well-timed.
But Crewe were guilty of some reckless defending and Anthony Tonkin handed Tudgay the chance to set up Rasiak for his second, but the striker lifted the ball over from eight yards.
However, when David Vaughan's loose pass presented the ball to Smith there was no reprieve and he tucked the ball to the right of Williams for the second.
Vaughan made amends though with a terrific 25-yard screamer into the top corner in the last minute to set up a grandstand finish in which Camp saved the day for George Burley's side when he blocked Jones' shot with his legs in stoppage time.
It allowed Derby to hold on to claim a vital three points - their first win in three games.